I haven’t seen a curriculum that develops quadratics quite in this way, but I’m having trouble giving up on this approach and going with anything else I’ve found. What do you think? Here’s how the unit would go:

FIRST TYPE OF EQUATION: $y = (x + a)(x+b)$

Step One: Learning to solve (x + a)(x + b) = c equations by treating them as multiplication equations. An important idea is that these equations can have 1, 2 or 0 solutions.

Step Two: In particular, learn how to efficiently solve (x + a)(x + b) = 0 and other similar, non-quadratic equations.

Step Three: Study these types of equations as functions. Check out what the zeroes represent in y = (x + a)(x + b).

(I’ve already done these activities in class while experimenting during the week before spring break.)

Step Four: Make generalizations about the graphs of these equations — about where the line of symmetry is, whether is curves up or curves down.

SECOND TYPE OF EQUATION: $y = x^2 + b$

Step Five: Check out a new type of equation $x^2 + b = c$ or $x^2 - b = c$. (I mean that $b$ is non-negative.) Learn to solve these equations by using what you already know about solving linear equations, with the new twist of taking roots of each side. And notice that sometimes these equations have 2, 1 or 0 solutions, and learn precisely what sorts of equations will have

Step Six: Graph these new equations, $y = x^2 + b$ or \$y = x^2 – b\$ especially in the case when $b$ is square. All that stuff above about lines of symmetry, zeroes, etc., study that but for these equations.

Step Seven: Big idea time. There are two equivalent ways of expressing many of these quadratic equations. No factoring, no multiplying binomials yet. Just notice: some of these $y = (x + a)(x + b)$ equations produce the same graphs as $y = x^2 + c$! (Mostly when $c$ is a square.) Let’s give arguments for why this is true, arguments about the zeroes, the lines of symmetry, and that these two equations share a vertex.

THIRD TYPE OF PROBLEM: MULTIPLYING AND FACTORING QUADRATICS

Step Eight: Learn to multiply binomials, like $(x + a)(x + b)$, and become equipped with a new algebraic way of doing the work of recognizing equivalent quadratic functions. Here we’ll especially focus on a difference of squares, $(x + a)(x - a)$.

Step Nine: Teach the rest of your quadratics unit at this point — including whatever other factoring you need to teach — while frequently asking the question “Will these equations produce the same graph or nah?”

***

This all seems to me a nice way to gradually build quadratics knowledge. If pushed on my design principles, I’d say that (a) I’m trying to be sensitive to the fact that the different types of equations that fall under ‘quadratics’ are of widely varying complexity and (b) I’m trying to make sure not to teach a connection between two mathematical objects before students have a chance to really become familiar with the different mathematical objects. (In other words, students would see lots of equations in factored/standard form before trying to connect them via multiplying or factoring one into the other.)

Is there any curriculum that structures a unit in a way that even roughly resembles this? I can’t develop too much of my own curricular stuff given my teaching load (four different courses: 3rd, 4th, 8th and Geometry) but I would love to try teaching this upcoming quadratics unit in something like this way.

Any materials or approaches you’ve seen for quadratics that resemble this? Can anyone talk me out of this approach? Where would I run into trouble, if I went against better judgement and developed my own materials for this unit?

## 3 thoughts on “An idea for teaching quadratics to 8th Graders”

1. I teach my quadratics unit in a somewhat similar way: We spend an entire unit analyzing quadratic functions in three forms (standard form, vertex form, factored form) and then spend the second unit making connections among those forms by distributing, factoring, completing the square, etc.

I really enjoy this approach because, as you said, kids get really familiar with each form before they do a lot of manipulating. In particular, they see how the a coefficient has the same effect in all forms, and they see how each form provides an easy way to find some of the important points of a parabola.

I believe I got this idea from the Carnegie Algebra 1 textbook that my district’s high school began using a couple of years ago. I didn’t use their lessons, but I believe I mirrored their overall sequence of ideas.

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2. 1). Factored form first. Yes! I especially love that you’re starting with factored form equations that are not set equal to 0. Building the conceptual awareness of the specialness of “zero” in the zero-product property. You might consider also emphasizing the importance of “product” in that phrase, by throwing in similar problems that look like factored form but have addition signs, like (x + 1) + (x – 5) = 0. Watch for students who set the (x+1) and (x-5) equal to zero independently.

2). I also love that you’re generalizing to cubic and higher-power equations instead of putting quadratics blinders on kids. Math is more fun when you punch through a wall and see a wide-open space, not a narrow tunnel, on the other side.

3). I love the Step 3 activities and will probably steal them this year. Unrelatedly, I love input-output tables laid out horizontally, so the values of the tables match up with the rough locations of the points on the graph (negative x-values on the left, and positive x-values on the right).

As part of Step 3, do you take some time and try to show kids what quadratic functions can actually model in real life? I use an adapted version of Will it Hit the Hoop, but even just a discussion might do.

As part of the discussion on what quadratics model, would you include the idea that they model situations in which the rate of change is changing at a constant rate? To me, that’s a big idea of quadratics that has to come out in the unit. Though I’m in Algebra 1. Not sure if that’s out of place in 8th grade. If you want resources on that, here are some:

— A quick Desmos activity I use: https://teacher.desmos.com/activitybuilder/custom/57693b4f0a66fa8a6a6e2284

— The larger lesson context of that Desmos activity: (though I’m not sure I still agree with everything in this blog post, it still describes the lesson fairly well)
https://ijkijkevin.wordpress.com/2016/03/24/creating-intellectual-need-for-multiplying-binomials/

–Here is the quadratics unit from the curriculum David Wees works on. You can see the big idea of changing-rate-of-change in the first big idea: https://curriculum.newvisions.org/math/course/algebra-i/quadratic-functions/

4). Step 4, love it.

5). In future years, I think it’s worth teaching this type of equation, and equations like 2(x+3)^2, earlier in the year, when you teach the “unwinding” strategy of equation-solving. The work kids have to do is really more connected to the order of operations than it is to quadratics. When I do this early in the year, we find the positive (or easiest) root only. If you did that, too, then your Step 5 would be simply teaching using the +- sign to get both roots, and also teaching about imaginary numbers/no solution. By the way, my decision to move the topic of unwinding quadratic equations to earlier in the year is 100% the result of my buying into your philosophy that students should get ample time to practice skills independently before having to make a connection between them.

6-9). Love them.

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